Problem: Find the distance between the point ${(-6, -5)}$ and the line $\enspace {y = -3x + 7}\thinspace$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
First, find the equation of the perpendicular line that passes through ${(-6, -5)}$ The slope of the blue line is ${-3}$ , and its negative reciprocal is ${\dfrac{1}{3}}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = \dfrac{1}{3}x + b}\thinspace$ We can plug our point, ${(-6, -5)}$ , into this equation to solve for ${b}$ , the y-intercept. $-5 = {\dfrac{1}{3}}(-6) + {b}$ $-5 = -2 + {b}$ $-5 + 2 = {b} = -3$ The equation of the perpendicular line is $\enspace {y = \dfrac{1}{3}x - 3}\thinspace$ We can see from the graph (or by setting the equations equal to one another) that the two lines intersect at the point ${(3, -2)}$ . Thus, the distance we're looking for is the distance between the two red points. The distance formula tells us that the distance between two points is equal to: $\sqrt{( x_{1} - x_{2} )^2 + ( y_{1} - y_{2} )^2}$ Plugging in our points ${(-6, -5)}$ and ${(3, -2)}$ gives us: $\sqrt{( {-6} - {3} )^2 + ( {-5} - {-2} )^2}$ $= \sqrt{( -9 )^2 + ( -3 )^2} = \sqrt{90} = 3\sqrt{10}$ The distance between the point ${(-6, -5)}$ and the line $\thinspace {y = -3x + 7}\enspace$ is $\thinspace3\sqrt{10}$.